Download Elastic Potential Energy

Chapter 7
Potential Energy
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7.1 Potential Energy

Potential energy is the energy
associated with the configuration of a
system of two or more interacting
objects or particles that exert forces on
each other
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The forces are internal to the system
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Types of Potential Energy
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There are many forms of potential
energy, including:
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Gravitational
Electromagnetic
Chemical
Nuclear
One form of energy in a system can be
converted into another
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System Example
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This system consists
of the Earth and a
book
An external force does
work on the system by
lifting the book through
Dy
The work done by the
external force on the
book is mgyb - mgya
Fig 7.1
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Gravitational Potential Energy
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Gravitational Potential Energy is
associated with an object at a given
distance above Earth’s surface
Assume the object is in equilibrium and
moving at constant velocity upwardly.
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The external force is F   mg
The displacement is Dr  ( yb  ya ) ĵ
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Gravitational Potential Energy,
cont
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The quantity mgy is identified as the
gravitational potential energy, Ug
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Ug = mgy
Units are joules (J)
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Gravitational Potential Energy,
final

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The gravitational potential energy depends
only on the vertical height of the object above
Earth’s surface
A reference configuration of the system must
be chosen so that the gravitational potential
energy at the reference configuration is set
equal to zero

The choice is arbitrary because the difference in
potential energy is independent of the choice of
reference configuration
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Potential Energy
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Similar as the work-kinetic energy theorem, the work
equals to the difference between the final and initial
values of some quantity of the system.
The quantity is called potential energy of the system.
Through the work, energy is transferred into the
system in a form different from kinetic energy.
The transferred energy is stored in the system.
The quantity Ug=mgy is called the gravitational
potential energy of the book-Earth system.
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7. 2 Conservation of Mechanical Energy
for an isolated system, example
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The isolated system is the book and the
Earth.
The work done on the book by the
gravitational force as it falls from some
height yb to a lower height ya
The motion of the book is free falling and
the kinetic energy of the book increases.
Won book = mgyb – mgya = -DUg
Won book = DKbook = Ka-Kb (Work-kinetic
theorem)
So, DK = -DUg Kb + Ugb = Ka + Uga
Fig 7.2
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Conservation of Mechanical
Energy for isolated systems

The mechanical energy of a system is the
algebraic sum of the kinetic and potential
energies in the system

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Emech = K + Ug
The Conservation of Mechanical Energy for
an isolated system is Kf + Ugf = Ki+ Ugi

An isolated system is one for which there are no
energy transfers across the boundary
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Fig 7.4
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Fig 7.5
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Elastic Potential Energy
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Elastic Potential Energy is associated
with a spring, Us = 1/2 k x2
The work done by an external applied
force on a spring-block system is
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W = 1/2 kxf2 – 1/2 kxi2
The work is equal to the difference
between the initial and final values of an
expression related to the configuration of
the system
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Elastic Potential Energy, cont
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This expression is the
elastic potential energy:
Us = 1/2 kx2
The elastic potential
energy can be thought
of as the energy stored
in the deformed spring
The stored potential
energy can be
converted into kinetic
energy
Fig 7.6
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Elastic Potential Energy, final

The elastic potential energy stored in a spring
is zero whenever the spring is not deformed
(U = 0 when x = 0)
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The energy is stored in the spring only when the
spring is stretched or compressed
The elastic potential energy is a maximum
when the spring has reached its maximum
extension or compression
The elastic potential energy is always positive

x2 will always be positive
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Conservation of Energy for
isolated systems

Including all the types of energy discussed so
far, Conservation of Energy can be
expressed as
DK + DU + DEint = DE system = 0 or
K + U + E int = constant
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K would include all objects
U would be all types of potential energy
The internal energy is the energy stored in a
system besides the kinetic and potential energies.
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7.3 Conservative Forces
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A conservative force is a force between
members of a system that causes no
transformation of mechanical energy to
internal energy within the system
The work done by a conservative force on a
particle moving between any two points is
independent of the path taken by the particle
The work done by a conservative force on a
particle moving through any closed path is
zero
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Nonconservative Forces

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A nonconservative force does not
satisfy the conditions of conservative
forces
Nonconservative forces acting in a
system cause a change in the
mechanical energy of the system
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Nonconservative Force,
Example
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Friction is an
example of a
nonconservative
force
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The work done
depends on the path
The red path will
take more work than
the blue path
Fig 7.7
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Mechanical Energy and
Nonconservative Forces

In general, if friction is acting in a
system:
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DEmech = DK + DU = -ƒkd
DU is the change in all forms of potential
energy
If friction is zero, this equation becomes
the same as Conservation of Mechanical
Energy
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Fig 7.8
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Nonconservative Forces,
Example 1 (Slide)
DEmech = DK + DU
DEmech =(Kf – Ki) +
(Uf – Ui)
DEmech = (Kf + Uf) –
(Ki + Ui)
DEmech = 1/2 mvf2 –
mgh = -ƒkd
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Nonconservative Forces,
Example 2 (Spring-Mass)
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Without friction, the
energy continues to be
transformed between
kinetic and elastic
potential energies and
the total energy remains
the same
If friction is present, the
energy decreases

DEmech = -ƒkd
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